On some Lr-biharmonic Euclidean Hypersurfaces

In decade eighty, Bang-Yen Chen introduced the concept of biharmonic hypersurface in the Euclidean space. An isometrically immersed hypersurface x : M → E is said to be biharmonic if ∆x = 0, where ∆ is the Laplace operator. We study the Lr-biharmonic hypersurfaces as a generalization of biharmonic ones, where Lr is the linearized operator of the (r + 1)th mean curvature of the hypersurface and in special case we have L0 = ∆. We prove that Lr-biharmonic hypersurface of Lr-finite type and also Lr-biharmonic hypersurface with at most two distinct principal curvatures in Euclidean spaces are r-minimal.